#6440774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,18)(17,21)(19,22)(20,26)(24,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,27,19,25,17,26)(14,20,21,24,22,28,16,23);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,22,19,21,18,14)(15,24,26,20,25,23,27,28);(2,4)(15,25)(20,28)(23,24)(26,27);(13,17,19,18)(14,21,22,16)(15,26,25,27)(20,24,28,23)

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307281, #72307338, #72307756
maximal normal subgroup of	#72307281, #72307338, #72307756

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347970);